Abstract

This paper develops the adjoint sensitivities to the free surface incompressible barotropic Euler equations in order to allow for the assimilation of measurements of currents and free surface elevations into an unsteady flow solution by boundary control. To calculate a variation in a surface variable, a mapping is used in the vertical to shift the problem into a fixed domain. Then a variation is evaluated from the Jacobian matrix of the mapping. After calculating a variation in the surface variable and applying the inverse transformation, the tangent linear model is considered in the original space where the adjoint equations are then derived. The method is demonstrated by application to an unsteady flow in an open channel (a 2D vertical section model). A wider application is to the construction of a fully three-dimensional coastal ocean model that allows assimilation of tidal elevation and current data.